Heluo Geometry in Scientific Applications - A Cross-Disciplinary Unifying Framework

This paper is the Scientific Applications Volume of the Heluo Geometry system. The mathematical foundations of Heluo Geometry are established in a separate paper — Discrete Sampling Geometry: A Rigorous Axiomatic Reconstruction (DOI: 10.5281/zenodo.20301581). That work starts from three axioms and rigorously proves the core structures: the uniqueness of the measurement function f(k) = cos(5πk/6) on ℤ/12ℤ, and the three-level quantization of Sheng-Ke curvature (Extremal / Transition / Saddle). This paper demonstrates the cross-scale projections of this geometric structure across five scientific domains. These mappings are not "analogies" or "fits," but mathematical necessities of discrete sampling geometry projecting onto different physical scales. Core Discoveries: Physics: From the discrete measurement function f(k) = cos(5πk/6), we derive the discrete analog of quantum probability amplitude; from the two-axis orthogonality, the Pauli matrices; from the three-level orbit structure, the gauge group SU(3)×SU(2)×U(1) with dimension decomposition 8+3+1=12; from discrete parameters, the fine-structure constant 1/α = 137.03599918. Chemistry: From the decreasing sequence of f(k), we derive the property variation across a row of the periodic table; f=0 (Saddle level) corresponds to the metalloid boundary; the extremal values correspond to alkali metals and noble gases. Biology: From the eight octants (2³ = 8) via Cartesian product, we obtain 64 combinations, corresponding to the 64 genetic codons. Cosmology: f=0 (Saddle level, free equilibrium) corresponds to dark energy; the 12-partition predicts CMB quadrupole-octupole alignment angles of 30° or 180°. Aerospace Engineering: Optimal orbital phase differences are integer multiples of 30°, confirmed by all existing GNSS constellations. Mathematical Foundation Statement: All geometric derivations in this paper are based on theorems from Discrete Sampling Geometry: A Rigorous Axiomatic Reconstruction (DOI: 10.5281/zenodo.20301581). That work proves from three axioms the uniqueness of the measurement function and the three-level quantization of Sheng-Ke curvature. This paper presents the projections and verifications of these theorems in specific scientific domains.